Derivative of (1+x²) to the tenth

Statement

Differentiate $f(x) = (1+x^2)^{10}$ with the circle diagram (two rings: $1+\square^2$, then $\triangle^{10}$).

Solution

Two rings: $x \to 1+x^2 \to (1+x^2)^{10}$. Partial derivatives: $2\square$ (for $1+\square^2$), $10\triangle^9$ (for $\triangle^{10}$). Multiplying and substituting ($\square=x$, $\triangle=1+x^2$): $f'(x) = 10(1+x^2)^9\cdot 2x = \boxed{20x(1+x^2)^9}.$

Links

Topics: Derivatives
Concepts: Composite derivative · Chain rule
Skills: Differentiating
Exercise type: Derivative computation